Solution of the Linear Boundary–Value Problem without Initial Conditions for the Second Order Hyperbolic Equation

Abstract

This paper studies boundary-value problem without initial conditions for the linear non-homogeneous second-order hyperbolic equation appearance \[u_{tt}-a^{2}u_{xx}=f(x,t),0\leq x\leq \pi ,0\leq t\leq T, u(0,t)=u(\pi ,t)=0, 0\leq t\leq T.\] Using the methods of the theory of differential equations in partial derivatives and methods of the theory of integral equations, for arbitrary functions \[\mu (z)\in C^{1}(\mathbf{R})\]  the exact solution of the indicated problem is constructed as \[u(x,t)=u^{0}(x,t)+\tilde{u}(x,t),\]   where \[u^{0}(x,t)= \frac{1}{2a}\int_{at-x}^{at+x}\mu (\alpha )d\alpha\] – the solution of the homogeneous equation and \[\tilde{u}(x,t)= \frac{1}{2a}\int_{0}^{t}d\tau \int_{x-a(t-\tau )}^{x+a(t-\tau)} f(\xi ,\tau )d\xi\] – a particular solution of the non-homogeneous equation. New existence conditions of the indicated problem are established. The classes of functions \[B_{0}^{-} =\left \{ \mu :\mu (z) =-\mu (-z)=\mu (\pi -z)\right \}, B^{-}=\left \{ f:f(x,t)=f(\pi -x,t) =-f(-x,t)\right \},\]  in which there is a classical solution of the linear boundary-value problem without initial conditions for the second order hyperbolic equations are discriminated. Based on the results operator A, which translates the class of functions \[B^{-}=\left \{ f:f(x,t)=f(\pi -x,t) =-f(-x,t)\right \}\]  in itself was built. This allows using it in the construction of approximate computations of the solution of boundary-value problems for the quasi-linear hyperbolic equations. The results are beginning of the boundary-value problems study without initial conditions for the second order hyperbolic equations in form \[u_{tt}-a^{2}u_{xx}=f(x,t,u_{t},u_{x}).\]  The proposed method of construction of the solution can be applied also to solve the semi-linear boundary-value problems.